The Apex Television Company has to decide on the number of $65$ and $55$ sets to be produced at one of its factories. Market research indicates that at most $280$ of the $65$ sets and $70$ of the $55$ sets can be sold per month. The maximum number of work$-$hours available is $3,800$ per month. A $65$ set requires $20$ work$-$hours and a $55$ set requires $10$ work$-$hours. Each $65$ set sold produces a profit of $$270$ and each $55$ set produces a profit of $$180.$ A wholesaler has agreed to purchase all the television sets produced if the numbers do not exceed the maxima indicated by the market research.

$1.$ Determine the number of $65$ and $55$ sets produced if the numbers do not exceed the maxima indicated by the market research.

$2.$ Determine the total profit.